Chapter 6
Probability
1
6.1: Chance Experiment & Sample
Space
A chance experiment is any activity or situation in
which there is uncertainty about which of two or more
possible outcomes will result (Not really a scientific
research experiment, but an experiment non the less…).
The collection of all possible outcomes of a chance
experiment is the sample space for the experiment.
2
Example
An experiment is to be performed to study student
preferences in the food line in the cafeteria. Specifically,
the staff wants to analyze the effect of the student’s
gender on the preferred food line (burger, salad or main
entrée).
3
1
Example - continued
The sample space consists of the following six possible
outcomes.
1. A male choosing the burger line.
2. A female choosing the burger line.
3. A male choosing the salad line.
4. A female choosing the salad line.
5. A male choosing the main entrée line.
6. A female choosing the main entrée line.
(If order is not important to the situation, could be said
burger line chosen by a male, etc…)
4
Example - continued
The sample space could be represented by using set
notation and ordered pairs.
sample space = {(male, burger), (female, burger), (male,
salad), (female, salad), (male, main entree), (female,
main entree)}
If we use M to stand for male, F for female, B for burger,
S for salad and E for main entrée the notation could be
simplified to
sample space = {MB, FB, MS, FS, ME, FE}
5
Example - continued
Yet another way of illustrating the sample
space would be using a picture called a “tree”
diagram.
Burger
Outcome (Male, Salad)
Salad
Main Entree
Male
Outcome (Female, Burger)
Burger
Female
Salad
Main Entree
This “tree” has two sets of “branches” corresponding to the two bits of
information gathered. To identify any particular outcome of the sample
space, you traverse the tree by first selecting a branch corresponding to
gender and then a branch corresponding to the choice of food line. (If order
is not important to the situation, burger, salad & entree could be 1st branch)
6
2
Events
An event is any collection of outcomes from
the sample space of a chance experiment.
A simple event is an event consisting of
exactly one outcome.
If we look at the lunch line example and use the following
sample space description {MB, FB, MS, FS, ME, FE}
The event that the student selected is male is given by
male = {MB, MS, ME}
The event that the preferred food line is the burger line is given
by
burger = {MB, FB}
The event that the person selected is a female that prefers the
salad line is {FS}. This is an example of a simple event &
there are 6 possible simple events that could occur).
7
Venn Diagrams
A Venn Diagram is an informal picture that
is used to identify relationships.
The collection of all possible outcomes of a
chance experiment are represented as the
interior of a rectangle.
The rectangle represents the
sample space and shaded
area represents the event A.
8
Forming New Events
Let A and B denote two events.
T h e e v e n t n o t A c o n s is ts o f a ll e x p e rim e n ta l
o u tc o m e s th a t a re n o t in e v e n t A . N o t A is
s o m e tim e s c a lle d th e c o m p le m e n t o f A a n d
is u s u a lly d e n o te d b y A c , A ’, C (A ), S - A , n o t
A , -A o r p o s s ib ly A .
The shaded area
represents the event
not A.
9
3
Forming New Events
Let A and B denote two events.
T h e e v e n t A o r B c o n s is ts o f a ll
e x p e rim e n ta l o u tc o m e s th a t a re in a t le a s t
o n e o f th e tw o e v e n ts , th a t is , in A o r in B
o r in b o th o f th e s e . A o r B is c a lle d th e
u n io n o f th e tw o e v e n ts a n d is d e n o te d b y
A ®B
A
∪ B.
The shaded area represents
the event A ∪ B.
10
Forming New Events
Let A and B denote two events.
T h e e v e n t A a n d B c o n s is ts o f a ll
e x p e rim e n ta l o u tc o m e s th a t a re in b o th o f
th e e v e n ts A a n d B . A a n d B is c a lle d th e
in te r s e c tio n o f th e tw o e v e n ts a n d is
d e n o te d b y A
A ­B
∩ B.
The shaded area represents
the event A ∩ B.
11
More on intersections
Two events that have no common outcomes are said to
be disjoint or mutually exclusive.
A and B are disjoint
events
12
4
More than 2 events
Let A1, A2, …, Ak denote k events
T h e e ve n ts A 1 o r A 2 o r … o r A k c o n s is t o f a ll
o u tc o m e s in a t le a s t o n e o f th e in d ivid u a l
® A2 A
®k] … ® A k ]
e ve n ts . [I.e
[i.e..,AA
1, 1A2, …,
T h e e ve n ts A 1 a n d A 2 a n d … a n d A k c o n s is t
o f a ll o u tc o m e s th a t a re s im u lta n e o u s ly in
e ve ry o n e o f th e in d ivid u a l e v e n ts .
[I.e .,A1A
[i.e.
,A
Ak] ­A k ]
1 ­A
2 ­…
2, …,
These k events are disjoint if no two of them have any
common outcomes.
13
Some illustrations
B
A
B
A
C
C
A, B & C are Disjoint
A∩B∩C
B
A
B
A
C
C
A∪B∪C
A ∩ B NOT C
14
Mutually Exclusive vs.
Independent
It’s common to confuse the concepts of ME and Indep.
If A happens, then event B cannot, or vice-versa. The
two events "it rained on Tuesday" and "it did not rain on
Tuesday" are mutually exclusive events. When
calculating the probabilities for MC events you add the
probabilities. With respect to independence, the
outcome of event A, has no effect on the outcome of
event B. Such as "It rained on Tuesday" and "My chair
broke at work". When calculating the probabilities for
independent events you multiply the probabilities. You
are effectively saying what is the chance of both events
happening bearing in mind that the two were unrelated.
15
5
Mutually Exclusive vs.
Independent cont…
So, if A and B are mutually exclusive, they cannot be
independent. If A and B are independent, they
cannot be mutually exclusive. However, If the events
were it rained today" and "I left my umbrella at
home" they are not mutually exclusive, but they are
probably not independent either, because one would
think that you'd be less likely to leave your umbrella
at home on days when it rains.
16
Mutually Exclusive example
What happens if we want to throw 1 and 6 in any order? This now
means that we do not mind if the first die is either 1 or 6, as we are
still in with a chance. But with the first die, if 1 falls uppermost,
clearly it rules out the possibility of 6 being uppermost, so the two
Outcomes, 1 and 6, are exclusive. One result directly affects the
other. In this case, the probability of throwing 1 or 6 with the first
die is the sum of the two probabilities, 1/6 + 1/6 = 1/3.
The probability of the second die being favorable is still 1/6 as the
second die can only be one specific number, a 6 if the first die is 1,
and vice versa.
Therefore the probability of throwing 1 and 6 in any order with one
die thrown twice is 1/3 x 1/6 = 1/18. Note that we multiplied the last
two probabilities as they were independent of each other!!!
17
Independent example
The probability of throwing a one & a six with two dice is
the result of throwing one with the first die and six with
the second die (or visa versa). The total possibilities are,
one from six outcomes for the first event and one from
six outcomes for the second, Therefore (1/6) * (1/6) =
1/36th or 2.77%. Since order didn’t matter (1,6 or 6,1) it’s
2/36th as there 2 ways to get it.
The two events are independent, since whatever
happens to the first die cannot affect the throw of the
second, the probabilities are therefore multiplied, and
remain 1/18th. Same P, but different way to calculate it.
Actually, this is the P(any pair) with 2 die.
18
6
6.2: Probability – Classical Approach
If a chance experiment has k
outcomes, all equally likely, then
each individual outcome has the
probability 1/k and the probability of
an event E is
P(E) =
number of outcomes favorable to E
number of outcomes in the sample space
19
Probability - Example
Consider the experiment consisting of rolling two
fair dice and observing the sum of the up faces. A
sample space description is given by
{(1, 1), (1, 2), … , (6, 6)}
where the pair (1, 2) means 1 is the up face of the
1st die and 2 is the up face of the 2nd die. This
sample space consists of 36 equally likely
outcomes.
Let E stand for the event that the sum is 6.
Event E is given by E={(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}.
The event consists of 5 outcomes, so
5
P(E) =
36
= 0.1389
20
Probability - Empirical Approach
Consider the chance experiment of rolling a
“fair” die. We would like to investigate the
probability of getting a “1” for the up face of the
die. The die was rolled and after each roll the up
face was recorded and then the proportion of
times that a 1 turned up was calculated and
Repeated Rolls of a Fair Die
plotted.
Proportion of 1's
0.200
0.180
1/6
Relative Frequency
0.160
0.140
0.120
0.100
0.080
0.060
0.040
0.020
0.000
0
21
200
400
600
800
1000
1200
1400
1600
1800
2000
# of Rolls
7
Probability - Empirical Approach
The process was simulated again and this time
the result were similar. Notice that the
proportion of 1’s seems to stabilize and in the
long run gets closer to the “theoretical” value of
1/6.
Repeated Rolls of a Fair Die
Proportion of 1's
0.350
Relative Frequency
0.300
0.250
0.200
1/6
0.150
0.100
0.050
0.000
0
200
400
600
800
1000
1200
1400
1600
1800
2000
# of Rolls
22
Probability - Empirical Approach
In many “real-life” processes and chance
experiments, the probability of a certain
outcome or event is unknown, but never the
less this probability can be estimated
reasonably well from observation. The
justification if the Law of Large Numbers.
Law of Large Numbers: As the number of
repetitions of a chance experiment increases, the
chance that the relative frequency of occurrence
for an event will differ from the true probability of
the event by more than any very small number
approaches zero.
23
Relative Frequency Approach
The probability of an event E, denoted by
P(E), is defined to be the value approached
by the relative frequency of occurrence of E
in a very long series of trials of a chance
experiment. Thus, if the number of trials is
quite large,
P (E ) ≈
n u m b e r o f tim e s E o c c u rs
n u m b e r o f t r ia ls
24
8
Methods for Determining Probability
1. The classical approach: Appropriate for experiments that
can be described with equally likely outcomes.
2. The subjective approach: Probabilities represent an
individual’s judgment based on facts combined with
personal evaluation of other information.
3. The relative frequency approach: An estimate is based
on an accumulation of experimental results. This estimate,
usually derived empirically, presumes a replicable chance
experiment.
25
6.3: Basic Properties of Probability
1. For any event E, 0≤P(E) ≤1.
2. If S is the sample space for an experiment,
P(S)=1.
3. If two events E and F are disjoint, then
P(E or F) = P(E) + P(F).
4. For any event E,
P(E) + P(not E) = 1 so,
P(not E) = 1 – P(E) and P(E) = 1 – P(not E).
26
Equally Likely Outcomes
Consider an experiment that can result in any
one of N possible outcomes. Denote the
corresponding simple events by O1, O2,… On.
If these simple events are equally likely to
occur, then
1
1
1
1. P(O1 ) = ,P(O2 ) = ,L,P(ON ) =
N
N
N
2. For any event E,
number of outcomes in E
P(E) =
N
27
9
Example
Consider the experiment consisting of randomly
picking a card from an ordinary deck of playing
cards (52 card deck).
Let A stand for the event that the card chosen is a
King.
The sample space is given by S =
{A♠, K♠,…,2♠, A♥ , K ♥ , …, 2♥, A♦,…, 2♦, A♣,…, 2♣}
and consists of 52 equally likely outcomes.
The event is given by
A={K♣, K♦, K♥, K♠}
and consists of 4 outcomes, so
P(A) =
28
4
1
= = 0.0769
52 13
Example
Consider the experiment consisting of rolling two fair dice
and observing the sum of the up faces.
Let E stand for the event that the sum is 7.
The sample space is given by
S={(1 ,1), (1, 2), … , (6, 6)}
and consists of 36 equally likely outcomes.
The event E is given by
E={(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
and consists of 6 outcomes, so
6
P(E) =
36
29
Example
Consider the experiment consisting of rolling two fair dice
and observing the sum of the up faces.
Let F stand for the event that the sum is 11.
The sample space is given by
S={(1 ,1), (1, 2), … , (6, 6)}
and consists of 36 equally likely outcomes.
The event F is given by
F={(5, 6), (6, 5)}
and consists of 2 outcomes, so
P(F) =
2
36
30
10
Addition Rule for Disjoint Events
Let E and F be two disjoint events.
One of the basic properties of probability is,
®F)=
P(E or F) = P(E ∪
F) = P(E)
P(E) ++ P(F)
P(F)
More Generally, if E1, E2 ,…,Ek are disjoint, then
®E
®…
®E
P(E1 or E2 or … or Ek) = P(E1 ∪
E2 ∪
…∪
Ek)
= P(E1) + P(E2) … + P(Ek)
31
Example
Consider the experiment consisting of rolling two
fair dice and observing the sum of the up faces.
Let E stand for the event that the sum is 7 and
F stand for the event that the sum is 11.
6
2
& P (F ) =
36
36
S in c e E a n d F a r e d is jo in t e v e n ts
P (E ) =
P (E U F ) = P (E ) + P (F ) =
6
2
8
+
=
36 36
36
32
A Leading Example
A study1 was performed to look at the
relationship between motion sickness and
seat position in a bus. The following table
summarizes the data.
Seat Position in Bus
Front Middle Back
Nausea
58
166
193
No Nausea 870
1163
806
1
“Motion Sickness in Public Road Transport: The Effect of Driver, Route and
Vehicle” (Ergonomics (1999): 1646 – 1664).
33
11
A Leading Example
Let’s use the symbols N, NC, F, M, B to
stand for the events Nausea, No Nausea,
Front, Middle and Back respectively.
Seat Position in Bus
Front Middle Back
Nausea
58
166
193
No Nausea 870
1163
806
Total
928
1329
999
Total
417
2839
3256
34
A Leading Example
Computing the probability that an individual
in the study gets nausea we have
P(N) =
417
= 0.128
3256
Seat Position in Bus
Front Middle Back
Nausea
58
166
193
No Nausea 870
1163
806
Total
928
1329
999
Total
417
2839
3256
35
A Leading Example
Other probabilities are easily calculated by
dividing the numbers in the cells by 3256 to
get
Seat Position in Bus
Front Middle Back
Nausea
0.018 0.051 0.059
No Nausea 0.267 0.357 0.248
Total
0.285 0.408 0.307
P(N and F)
P(F)
P(N)
Total
0.128
0.872
1.000
P(M and NC)
36
12
A Leading Example
The event “a person got nausea given
he/she sat in the front seat” is an example of
what is called a conditional probability.
Of the 928 people who sat in the front, 58 got
nausea so the probability that “a person got nausea
given he/she sat in the front seat is
58
= 0 .0 6 2 5
928
37
6.4: Conditional Probability
If we want to see if nausea is related to seat
position we might want to calculate the probability
that “a person got nausea given he/she sat in the
front seat.”
We usually indicate such a conditional probability
with the notation P(N | F).
P(N | F) stands for the “probability of N given F.
38
Conditional Probability
Let E and F be two events with P(F) > 0.
The conditional probability of the event
E given that the event F has occurred,
denoted by P(E|F), is
P (E | F ) =
P (E I F )
P (F )
39
13
Example
A survey of job satisfaction2 of teachers was
taken, giving the following results
L
E
V
E
L
2
Job Satisfaction
Satisfied Unsatisfied Total
College
74
43
117
High School
224
171
395
Elementary
126
140
266
Total
424
354
778
“Psychology of the Scientist: Work Related Attitudes of U.S. Scientists”
(Psychological Reports (1991): 443 – 450).
40
Example
If all the cells are divided by the total number
surveyed, 778, the resulting table is a table of
empirically derived probabilities.
L
E
V
E
L
Job Satisfaction
Satisfied Unsatisfied Total
College
0.095
0.055
0.150
High School
0.288
0.220
0.508
Elementary
0.162
0.180
0.342
Total
0.545
0.455
1.000
41
Example
L
E
V
E
L
Job Satisfaction
Satisfied Unsatisfied Total
0.095
0.055
0.150
High School 0.288
0.220
0.508
Elementary
0.162
0.180
0.342
Total
0.545
0.455
1.000
College
For convenience, let C stand for the event that
the teacher teaches college, S stand for the
teacher being satisfied and so on. Let’s look at
some probabilities and what they mean.
is the proportion of teachers who are
college teachers.
is the proportion of teachers who are
P(S) = 0.545
satisfied with their job.
P(C I S) = 0.095 is the proportion of teachers who are
college teachers and who are satisfied
with their job.
P(C) = 0.150
42
14
Example
Job Satisfaction
Satisfied Unsatisfied Total
0.095
0.055
0.150
High School 0.288
0.220
0.508
Elementary
0.162
0.180
0.342
Total
0.545
0.455
1.000
College
L
E
V
E
L
The proportion of teachers who are
college teachers given they are satisfied is
P (C | S ) =
P (C I S )
0 .0 9 5
=
= 0 .1 7 5
P (S )
0 .5 4 5
Restated: This is the proportion of satisfied that are
college teachers.
43
Example
L
E
V
E
L
Job Satisfaction
Satisfied Unsatisfied Total
College
0.095
0.055
0.150
High School 0.288
0.220
0.508
Elementary
0.162
0.180
0.342
Total
0.545
0.455
1.000
The proportion of teachers who are
satisfied given they are college teachers is
P (S I C )
P (C I S )
=
P (C )
P (C )
0 .0 9 5
=
= 0 .6 3 2
0 .1 5 0
P (S | C ) =
Restated: This is the proportion of college teachers
that are satisfied.
44
6.5: Independence
Two events E and F are said to be independent if
P(E|F)=P(E).
If E and F are not independent, they are said to be
dependent events.
If P(E|F) = P(E), it is also true that P(F|E) = P(F) and vice
versa. So if E is independent of F, F is independent of
E.
45
15
Example
L
E
V
E
L
Job Satisfaction
Satisfied Unsatisfied Total
College
0.095
0.055
0.150
High School 0.288
0.220
0.508
Elementary
0.162
0.180
0.342
Total
0.545
0.455
1.000
P(C) = 0.150 and P(C | S) =
P(C I S) 0.095
=
= 0.175
P(S)
0.545
P(C|S) ≠ P(C) so C and S are dependent events.
46
Multiplication Rule for Independent Events
The events E and F are independent if and
only if P(E ∩ F) = P(E)P(F)
That is, independence implies the relation
P(E ∩ F) = P(E)P(F), and this relation implies
independence.
47
Example
Consider the person who purchases from two
different manufacturers a TV and a VCR.
Suppose we define the events A and B by
A = event the TV doesn’t work properly
B = event the VCR doesn’t work properly
Suppose P(A) = 0.01 and P(B) = 0.02.
If we assume that the events A and B are
independent3, then
P(A
∩B)
B)==(0.01)(0.02)
(0.01)(0.02)==0.0002
0.0002
P(A and
and B)
B) == P(A
P(A­
3
This assumption seems to be a reasonable assumption since the
manufacturers are different.
48
16
Example
Consider the teacher satisfaction survey
L
E
V
E
L
Job Satisfaction
Satisfied Unsatisfied
College
0.095
0.055
High School
0.288
0.220
Elementary
0.162
0.180
Total
0.545
0.455
Total
0.150
0.508
0.658
1.000
P(C) = 0.150, P(S) = 0.545 and P(C ∩ S) = 0.095
Since P(C)P(S) = (0.150)(0.545) = 0.08175 and
P(C ∩ S) = 0.095, P(C ∩ S) ≠ P(C)P(S). This
shows that C & S are dependent events.
49
Sampling Schemes
Sampling is with replacement if, once selected, an
individual or object is put back into the population before
the next selection.
Sampling is without replacement if, once selected, an
individual or object is not returned to the population prior
to subsequent selections.
50
Example
Suppose we are going to select three cards from an
ordinary deck of cards. Consider the events:
E1 = event that the first card is a king
E2 = event that the second card is a king
E3 = event that the third card is a king.
51
17
Example – With Replacement
If we select the first card and then place it back in
the deck before we select the second, and so on,
the sampling will be with replacement.
P(E1 ) = P(E 2 ) = P(E3 ) =
4
52
P(E1 I E2 I E3 ) = P(E1 )P(E2 )P(E3 )
=
4 4 4
= 0.000455
52 52 52
52
Example – Without Replacement
If we select the cards in the usual manner
without replacing them in the deck, the
sampling will be without replacement.
P(E1 ) =
4
3
2
, P(E2 ) = , P(E3 ) =
52
51
50
P(E1 I E2 I E3 ) = P(E1 )P(E2 )P(E3 )
=
4 3 2
= 0.000181
52 51 50
53
A Practical Example
Suppose a jury pool in a city contains 12000
potential jurors and 3000 of them are black
women. Consider the events
E1 = event that the first juror selected is a black
woman
E2 = event that the second juror selected is a
black woman
E3 = event that the third juror selected is a black
woman
E4 = event that the forth juror selected is a black
woman
54
18
A Practical Example
Clearly the sampling will be without replacement so
3000
2999
, P(E2 ) =
,
12000
11999
2998
2997
P(E3 ) =
,P(E 4 ) =
11998
11997
P(E1 ) =
So P(E1 I E2 I E3 I E4 )
=
3000 2999 2998 2997
= 0.003900
12000 11999 11998 11997
55
A Practical Example - continued
If we “treat” the Events E1, E2, E3 and E4 as being with
replacement (independent) we would get
P(E1 ) = P(E 2 ) = P(E3 ) =
3000
= 0.25
12000
So P(E1 I E2 I E3 I E4 ) = (0.25)(0.25)(0.25)(0.25)
= 0.003906
56
A Practical Example - continued
Notice the result calculate by sampling without
replacement is 0.003900 and the result calculated by
sampling with replacement is 0.003906. These results
are substantially the same.
Clearly when the number of items is large and the
number in the sample is small, the two methods give
essentially the same result.
57
19
An Observation
If a random sample of size n is taken from a
population of size N, the theoretical probabilities of
successive selections calculated on the basis of
sampling with replacement and on the basis of
sample without replacement differ by insignificant
amounts under suitable circumstances.
Typically independence is assumed for the purposes
of calculating probabilities when the sample size n is
less than 5% of the population size N.
58
6.6: General Addition Rule for Two
Events
For any two events E and F,
P(E U F) = P(E) + P(F) − P(E I F)
59
Example
Consider the teacher satisfaction survey
L
E
V
E
L
College
High School
Elementary
Total
Job Satisfaction
Satisfied Unsatisfied Total
0.095
0.055
0.150
0.288
0.220
0.508
0.162
0.180
0.658
0.545
0.455
1.000
P(C) = 0.150, P(S) = 0.545 and
P(C ∩ S) = 0.095, so
P(C ∪ S) = P(C) + P(S) – P(C ∩ S)
= 0.150 + 0.545 – 0.095
= 0.600
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20
General Multiplication Rule
For any two events E and F,
P(E I F) = P(E | F)P(F)
From symmetry we also have
P(E I F) = P(F | E)P(E)
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Example
18% of all employees in a large company are
secretaries and furthermore, 35% of the
secretaries are male. If an employee from this
company is randomly selected, what is the
probability the employee will be a secretary
and also male.
Let E stand for the event the employee is male.
Let F stand for the event the employee is a secretary.
The question can be answered by noting that
P(F) = 0.18 and P(E|F) = 0.35 so
P(E I F) = P(E | F)P(F) = (0.35)(0.18) = 0.063
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Bayes Rule
If B1 and B2 are disjoint events with
P(B1) + P(B2) = 1, then for any event E
P(B1|E) =
P(E|B1)P(B1)
P(E|B1)P(B1)+P(E|B2)P(B2)
More generally, if B1, B2, …, Bk are disjoint events with
P(B1) + P(B2) + … P(Bk) = 1, then for any event E
P(Bi | E) =
P(Ei)P(B
| Bi ) i)
P(E|B
P(E | B1 )P(B1 ) + P(E | B2 )P(B2 ) + L + P(E | Bk )P(Bk )
Use when given P(E|B1) & you want P(B1|E)
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21
Example
A company that makes radios, uses three different
subcontractors (A, B and C) to supply on switches used
in assembling a radio. 50% of the switches come from
company A, 35% of the switches come from company B
and 15% of the switches come from company C.
Furthermore, it is known that 1% of the switches that
company A supplies are defective, 2% of the switches
that company B supplies are defective and 5% of the
switches that company C supplies are defective.
Thus, we know the defective rate given a switch is from
a specific company.
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Example
If a radio from this company was inspected and failed
the inspection because of a defective on switch, what
are the probabilities that that switch came from each of
the suppliers.
So we want to find the probability it came from a
specific company given the switch is defective. This is
the opposite of what we were given, so we use Bayes
Rule.
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Example - continued
Define the events
S1 = event that the on switch came from subcontractor A
S2 = event that the on switch came from subcontractor B
S3 = event that the on switch came from subcontractor C
D = event the on switch was defective
From the problem statement we have
P(S1) = 0.5, P(S2) = 0.35 , P(S3) = 0.15
P(D|S1) =0.01, P(D|S2) =0.02, P(D|S3) =0.05
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22
Example - continued
P(Switch came from supplier A given it was
defective) =
P(S 1 | D) =
P(D
| S 11))
P(D|S
1)P(S
P(D | S 1 )P(S 1 ) + P(D | S 2 )P(S 2 ) + P(D | S 3 )P(S 3 )
(.5)(.01)
(.5)(.01) + (.35)(.02) + (.15)(.05)
.005
.005
=
=
= .256
.005 + .007 + .0075 .0195
=
Similarly, P(S2|D) =
(.35)(.02)
,. P(S2|D) = .359
(.5)(.01)+(.35)(.02)+(.15)(.05)
P(S3|D) =
(.15)(.05)
,. P(S3|D) = .385
(.5)(.01)+(.35)(.02)+(.15)(.05)
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Example - continued
These calculations show that 25.6% of the
defective switches are supplied by subcontractor
A, 35.9% of the defective on switches are
supplied by subcontractor B and 38.5% of the
defective on switches are supplied by
subcontractor C.
Even though subcontractor C supplies only a small
proportion (15%) of the switches, it supplies a
reasonably large proportion of the defective
switches (38.9%).
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6.7: Estimating Probabilities
Empirically & Using Simulation
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23
Estimating Probability Empirically
Common to use observed long term proportions to
estimate probabilities empirically.
• Observe large # of chance outcomes in a
controlled environment
• Using your knowledge of long run relative
frequencies & the law of large numbers
estimate the probability of the observed event.
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Example
Men & women frequently express intimacy via touch.
Holding hands is an example. Some researchers
say not only does this act indicate intimacy, but may
also indicate status differences.
Research indicates that the males predominately
assume the overhand status, women the
underhand… The authors of “Men & Women Holding
Hands: Whose Hand is Uppermost” believe height
may be the more reasonable explanation.
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Example…
Number of hand holding couples
Gender of Person with uppermost hand
Men
Women
Total
Man taller
2149
299
2448
Equal Ht
780
246
1026
Woman taller
241
205
446
3170
750
3920
Total
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24
Example
Assuming that the reported hand holding couples are
representative of the population of hand holding couples,
we can estimate various probabilities, i.e
P(man’s uppermost) = 3170/3920 = 0.809
P(woman’s uppermost) = 750/3920 = 0.191
P(man taller uppermost) = 2149/2448 = 0.878 conditional
P(woman taller uppermost) = 205/446 = 0.460 conditional
So men still have the upperhand
Note: P(man taller uppermost) ≠ P(man’s uppermost),
therefore NOT independent events
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Estimating Probability Simulation
When impractical to measure empirically or can’t
measure analytically..
• Design method that uses a random mechanism
to represent an observation.
• Generate an obs using your method &
determine if the outcome of interest has
occurred. Repeat a large # of times.
• Calculate the estimated probability by dividing
the # of obs for which the outcome of interest
occurred by the total # of obs
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Example
Lets do a simulation for the multiple choice portion of a 20
question test with 5 choices for each question (only 1 being
correct). Using a random # table we randomly pick a place to
start & get the following #’s (note take 20 #’s at a time since there are 20 questions).
Since there are 5 choices & 10 possibilities for each of the 20
digits we need 2 #’s to represent success for each of the 20
digits (e.g. Lets let 0 & 1 be success)
Test #1: 9 4 6 0 6 9 7 8 8 2 5 2 9 6 0 1 4 6 0 5 4 correct
Test #2: 6 6 9 5 7 4 4 6 3 2 0 6 0 8 9 1 3 6 1 8 4 correct
Test #3: 0 7 1 7 7 2 9 5 4 8 6 2 7 5 1 0 4 3 0 7 5 correct
etc…
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